Abstract

We study Bayesian inference in statistical linear inverse problems with Gaussian noise and priors in a separable Hilbert space setting. We focus our interest on the posterior contraction rate in the small noise limit, under the frequentist assumption that there exists a fixed data-generating value of the unknown. In this Gaussian-conjugate setting, it is convenient to work with the concept of squared posterior contraction (SPC), which is known to upper bound the posterior contraction rate. We use abstract tools from regularization theory, which enable a unified approach to bounding SPC. We review and re-derive several existing results, and establish minimax contraction rates in cases which have not been considered until now. Existing results suffer from a certain saturation phenomenon, when the data-generating element is too smooth compared to the smoothness inherent in the prior. We show how to overcome this saturation in an empirical Bayesian framework by using a non-centered data-dependent prior.

which proves the first assertion. The variance is \(\mathbb E^{x^{\ast }} \left \| x^\delta _\alpha - \mathbb E^{x^{\ast }}x^\delta _\alpha \right \|{ }_{ }^{2}\), and this can be written as in (8), by using similar reasoning as for the bias term.

Since C0 has finite trace, it is compact, and we use the eigenbasis (arranged by decreasing eigenvalues) uj, j = 1, 2, … Under Assumption 1 this is also the eigenbasis for T∗T. If tj, j = 1, 2, … denote the eigenvalues then we see that

For the first item (1), we notice that \(\varphi \prec \varTheta _{\psi }^{2}\) if and only if φ(f2(t)) ≺ t. The linear function t↦t is a qualification of Tikhonov regularization with constant γ = 1. Thus, by Lemma 3 we have

where the right hand side is positive since the operator C0 is positive definite. Thus, if α < α′ then \(S_{T,C_0}(\alpha ) - S_{T,C_0}(\alpha ^{\prime })\) is positive, which proves the first assertion.

The proof of the second assertion is simple, and hence omitted. To prove the last assertion we use the partial ordering of self-adjoint operators in Hilbert space, that is, we write A ≤ B if 〈Ax, x〉≤〈Bx, x〉, x ∈ X, for two self-adjoint operators A and B. Plainly, with \(a:= \left \| T^{\ast }T \right \|{ }_{}\), we have that T∗T ≤ aI. Multiplying from the left and right by \(C_0^{1/2}\) this yields B∗B ≤ aC0, and thus for any α > 0 that αI + B∗B ≤ αI + aC0. The function t↦ − 1/t, t > 0 is operator monotone, which gives \(\left ( \alpha I + aC_0 \right )^{-1} \leq \left ( \alpha I + B^{\ast }B \right )^{-1}\). Multiplying from the left and right by \(C_0^{1/2}\) again, we arrive at

If \(S_{T,C_0}(\alpha )\) were uniformly bounded from above, then there would exist a finite natural number, say N, such that \(t_{N} \geq \frac \alpha a > t_{N+1}\), for α > 0 small enough. But this would imply that tN+1 = 0, which contradicts the assumption that C0 is positive definite.

In this example the explicit solution of Eq. (16) in Theorem 1 is more difficult. However, as discussed in Sect. 3.4, it suffices to asymptotically balance the squared bias and the posterior spread using an appropriate parameter choice α = α(δ). Indeed, under the stated choice of α the squared bias is of order

According to the considerations in Remark 10, it is straightforward to check that without preconditioning the best SPC rate that can be established is \(\delta ^{\frac {4+8a+8p}{3+4a+6p}}\) which proves item (1). In the preconditioned case, the explicit solution of Eq. (16) in Theorem 1, which in this case has the form

is again difficult. However, as discussed in Sect. 3.4, it suffices to asymptotically balance the squared bias and the posterior spread using an appropriate parameter choice α = α(δ). Indeed, using [3, Lem 4.5] we have that the solution to the above equation behaves asymptotically as the stated choice of α, and substitution gives the claimed rate.

is difficult. As discussed in Sect. 3.4, it suffices to asymptotically balance the squared bias and the posterior spread using an appropriate parameter choice α = α(δ). Indeed, under the stated choice of α both quantities are bounded from above by \(\delta ^{\frac {2\beta }{\beta +q}}\). For item (2), according to the considerations in Remark 10, it is straightforward to check that without preconditioning the best SPC rate that can be established is \(\delta ^{\frac {4q}{\beta +q}}\).